⚫ Let p be the proportion of slips of paper that say "I have used illegal drugs". p is chosen by the researcher in advance. Each participant chooses a random slip of paper from the hat and answers "yes" or "no" to whether the statement on that slip is true. The slip is then returned to the hat. The researcher doesn't know which type of slip the participant had. • Let y be the probability that a participant will say "yes", ⚫ and d be the probability that a participant has used illegal drugs. 1. Find y in terms of d and p. 2. Given that the researcher is interested in the true proportion of people using illegal drugs, d, what would be the worst possible choice of p that the researcher could make in designing the survey? Explain. Now consider the following alternative system. Suppose that proportion p of the slips of paper say "I have used illegal drugs", but that now the remaining (1 - p) say "I was born in winter" rather than "I have not used illegal drugs". Assume that 1/4 of people are born in winter, and that a person's season of birth is independent of whether they have used illegal drugs. 3. Find d, in terms of y and p. 4. A randomized response survey like in part (3.) is conducted with 100 participants and p = ½. Each participant draws a slip of paper from a box (with replacement) and answers the question on the paper truthfully. At the end of the survey, the responses are counted and there are 20 "yes" responses and 80 "no" responses.
⚫ Let p be the proportion of slips of paper that say "I have used illegal drugs". p is chosen by the researcher in advance. Each participant chooses a random slip of paper from the hat and answers "yes" or "no" to whether the statement on that slip is true. The slip is then returned to the hat. The researcher doesn't know which type of slip the participant had. • Let y be the probability that a participant will say "yes", ⚫ and d be the probability that a participant has used illegal drugs. 1. Find y in terms of d and p. 2. Given that the researcher is interested in the true proportion of people using illegal drugs, d, what would be the worst possible choice of p that the researcher could make in designing the survey? Explain. Now consider the following alternative system. Suppose that proportion p of the slips of paper say "I have used illegal drugs", but that now the remaining (1 - p) say "I was born in winter" rather than "I have not used illegal drugs". Assume that 1/4 of people are born in winter, and that a person's season of birth is independent of whether they have used illegal drugs. 3. Find d, in terms of y and p. 4. A randomized response survey like in part (3.) is conducted with 100 participants and p = ½. Each participant draws a slip of paper from a box (with replacement) and answers the question on the paper truthfully. At the end of the survey, the responses are counted and there are 20 "yes" responses and 80 "no" responses.
College Algebra
7th Edition
ISBN:9781305115545
Author:James Stewart, Lothar Redlin, Saleem Watson
Publisher:James Stewart, Lothar Redlin, Saleem Watson
Chapter9: Counting And Probability
Section9.4: Expected Value
Problem 20E
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Question
Solve by hand and show the steps, explinations, and calculations. Solve all parts.
Transcribed Image Text:⚫ Let p be the proportion of slips of paper that say "I have used illegal drugs". p is chosen
by the researcher in advance.
Each participant chooses a random slip of paper from the hat and answers "yes" or "no" to
whether the statement on that slip is true. The slip is then returned to the hat. The researcher
doesn't know which type of slip the participant had.
• Let y be the probability that a participant will say "yes",
⚫ and d be the probability that a participant has used illegal drugs.
1. Find y in terms of d and p.
2. Given that the researcher is interested in the true proportion of people using illegal
drugs, d, what would be the worst possible choice of p that the researcher could make in
designing the survey? Explain.
Now consider the following alternative system. Suppose that proportion p of the slips of
paper say "I have used illegal drugs", but that now the remaining (1 - p) say "I was born in
winter" rather than "I have not used illegal drugs". Assume that 1/4 of people are born in
winter, and that a person's season of birth is independent of whether they have used illegal
drugs.
3. Find d, in terms of y and p.
4. A randomized response survey like in part (3.) is conducted with 100 participants and
p = ½.
Each participant draws a slip of paper from a box (with replacement) and answers the
question on the paper truthfully. At the end of the survey, the responses are counted
and there are 20 "yes" responses and 80 "no" responses.
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